Venn Diagrams

The part common to both circles contains exactly those elements that belong to A and also belong to B. This set is called the intersection and is written as \(A\cap B\). In contrast, \(A\cup B\) includes every element in either circle, while \(A-B\) contains elements only in A.

The union A ∪ B contains every element lying in at least one of the two circles. Therefore, the region outside both circles but still inside the universal set contains all elements that are not in A ∪ B. This is the complement of the union, written as (A ∪ B)′. Hence, option C is correct.

The governing idea is set inclusion. A ⊆ B states that every element of A is also an element of B. In a Venn diagram, this is represented by drawing the complete region for A inside the region for B. A may be a smaller set, but none of its elements can lie outside B. Option B contradicts inclusion, option C is impossible because sets lie within U, and option D is outside A itself. Thus, option A is correct.

If A and B have no common element, their intersection is empty: \(A\cap B=\varnothing\). Such sets are called disjoint sets. In a Venn diagram, their circles are drawn separately without overlap, while both circles remain inside the universal-set rectangle U.

The union A∪B contains every element that lies in A, in B, or in their intersection. The three stated Venn-diagram regions are disjoint, so each must be counted exactly once: n(A∪B)=12+5+8=25. Therefore option C is correct. Adding only the exclusive regions would give 20 and would wrongly omit the five common elements.

The elements in neither A nor B are the elements in the universal set that lie outside the union \(A\cup B\). Therefore, this region is the complement of the union, \((A\cup B)'\), and its cardinality is \(n(U)-n(A\cup B)=52-37=15\). Thus, option A is correct. The value 37 counts the union itself, 52 counts the entire universal set, and 89 incorrectly adds the two given numbers.

The notation A∩B′ means an element must be in A and also in the complement of B. Being in B′ means not being in B, so the expression describes the A-only region. Thus option A is correct and is equivalent to A−B. The common region fails because it lies inside B, while the B-only and outside regions are not in A.

De Morgan’s first law states that the complement of an intersection is the union of the complements: (A ∩ B)' = A' ∪ B'. An element is outside A ∩ B if it is absent from at least one of A or B. Therefore, the regions outside the common overlap are included. Option B represents the complement of a union, not of an intersection.

A two-set Venn diagram partitions U into four disjoint regions: outside both sets, only A, only B, and A ∩ B. Let the unknown intersection size be x. Then 15 + 25 + 18 + x = 80, so x = 80 − 58 = 22. Therefore n(A ∩ B) = 22. The other choices result from omitting or combining one or more regions incorrectly.

The union consists of three disjoint Venn-diagram regions: elements only in A, elements common to A and B, and elements only in B. Thus, 44 = 16 + n(A ∩ B) + 19. Subtracting the two exclusive regions gives n(A ∩ B) = 44 − 16 − 19 = 9. Hence, option A is correct. Options B and C are the given exclusive counts, while 35 is their combined total.

Let T be the set of people who drink tea and M the set of people who drink milk. The 17 tea drinkers include the 6 people who drink both beverages. Therefore, the number who drink only tea is n(T) − n(T ∩ M) = 17 − 6 = 11. The total group size and milk count are not needed for this calculation, so option B is correct.

The total of 22 swimming participants includes the 8 students who participate in both running and swimming. To isolate the swimming-only region, subtract the overlap: n(only swimming) = n(swimming) − n(running ∩ swimming) = 22 − 8 = 14. Thus, 14 students participate only in swimming.

The central region of a three-set Venn diagram contains elements that lie inside A, inside B, and inside C simultaneously. The intersection symbol ∩ means “common to all selected sets,” so the central region is represented by A ∩ B ∩ C. A union includes elements in any set, while A − B and C′ describe different regions.

The elements only in A are those in A but not in B, so their number is n(A)−n(A∩B)=11−4=7. Similarly, the elements only in B number n(B)−n(A∩B)=9−4=5. Therefore, the requested total is 7+5=12, so option B is correct. Notice that 16 is n(A∪B), which includes the common four elements and therefore does not answer the question about exclusive regions.

The condition n(A∩B)=0 means that A and B have no common element. In a Venn diagram, their circles therefore do not overlap; such sets are called disjoint or mutually exclusive. They cannot be equal because their cardinalities are different. Neither set is necessarily a subset or a complement based only on the information given. Hence option A is correct.

For two finite sets, the inclusion–exclusion principle gives n(A ∪ B) = n(A) + n(B) − n(A ∩ B). The subtraction is necessary because the 8 elements common to A and B are counted once in n(A) and once in n(B). Thus n(A ∪ B) = 26 + 18 − 8 = 36. Therefore, option A is correct; 44 results from forgetting the overlap.

Elements that are neither in A nor in B lie outside the union A ∪ B, so they form the complement (A ∪ B)′ within the universal set U. The universal set is partitioned into the union and its complement. Therefore n((A ∪ B)′) = n(U) − n(A ∪ B) = 58 − 41 = 17. Option A is correct; 41 is the union itself and 58 is the total universe.

The complement A′ is defined relative to the universal set U. It consists of all elements that belong to U but do not belong to A. On a Venn diagram, this is the entire part of the rectangle outside circle A. Thus the correct representation is A′, option B.

The union A ∪ B includes every element belonging to A, to B, or to both. Thus the two circles, including their overlap, are shaded. The only region left unshaded inside U is outside both circles, which is the complement (A ∪ B)'. Hence option D is correct.

The middle overlapping region of two circles contains exactly those elements that belong to both A and B. This set is called their intersection and is written as A ∩ B. The union includes both circles, while A − B includes only the non-overlapping part of A. Therefore, option C is correct.

The union \(A\cup B\) contains every element that belongs to at least one of the two sets. The elements in neither set are outside this union but still inside the universal set. Hence their number is \(n(U)-n(A\cup B)=85-56=29\). Therefore, option A is correct. The value 56 counts the union itself, not the region outside both sets.

The union contains every element in at least one of the sets. Its three disjoint Venn-diagram regions are only A, the intersection, and only B. Therefore, \(n(A\cup B)=18+7+14=39\). Each element is counted exactly once because the regions are disjoint. Hence option C is correct; adding only the two exclusive regions would incorrectly omit the seven common elements.

The three listed regions inside \(A\cup B\) are disjoint, so their total is \(26+15+31=72\). The universal set has 94 elements, and the remaining elements are outside both sets. Hence, the number in neither set is \(94-72=22\). Option A is correct; 72 is the union count, not the outside count.

The universal set is divided into four disjoint regions: only A, the intersection, only B, and neither set. Their total is 76. Therefore, \(n(A\cap B)=76-(20+23+18)=76-61=15\). Option A is correct. The value 61 is only the sum of the three known regions, so it excludes the unknown intersection rather than representing it.

A two-set Venn diagram divides A ∪ B into three disjoint regions: only A, A ∩ B, and only B. Their counts must add to the union total. Therefore 63 = 24 + n(A ∩ B) + 27, so n(A ∩ B) = 63 − 24 − 27 = 12. Option A is correct. The numbers 24 and 27 are separate-only regions, not the common region.